By Angel

ISBN-10: 0120579502

ISBN-13: 9780120579501

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**Extra info for Dynamic Programming and Partial Differential Equations**

**Example text**

Much of what follows in the succeeding chapters will be based on matrix versions of these results. EXERCISES 1. Prove (9). 2. Find the recurrence equations for the coefficients r k , s, and t k . Use these results to obtain a recurrence equation for the x k . BlBLlOG RAPHY AND COMMENT See R. Bellman, Introduction to the Mathematical Theory of Control Processes I : Linear Equations and Quadratic Criteria, Academic Press, New York, 1967. Section 7 . See R. Bellman, “Functional Equations in the Theory of Dynamic Programming XVII : Minimum Convolutions and Green’s Functions,” J.

2). The solution of linear differential equations of this nature can be effected by means of the solution of M-dimensional linear algebraic systems. The associated variational problem is the minimization of The positive-definite nature of this functional ensures the existence and uniqueness of the solution of (1) and (2). 14. Irregular Grid There is no necessity, of course, to employ a regular grid. ,Yn-l,Yn (1) 9 where 0 = y o < y1 < y 2 < ... ,yn. (2) One way to determine the quantities ivi and yi is to require that the formula be exact for polynomials of degree n or less.

This transformation, of course, is the beginning of the real problem, that o f obtaining useful analytic and computational results from the linear system. 8. Rectangular Region Let us see what this involves for a rectangular region (Fig. 3). Let integers M and N be chosen and the positive quantities A and 6 be determined by M A = a, N 6 = 6. , N . q(rnd, h ) . 'obr7pl We suppose t h a t g ( s , y) is continuous so that g(0, 0) and g ( m A , N 6 ) are unambiguously defined by the foregoing. 01 29 10.

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